baroco

Description: "Baroco", one of the syllogisms of Aristotelian logic. All ph is ps , and some ch is not ps , therefore some ch is not ph . In Aristotelian notation, AOO-2: PaM and SoM therefore SoP. For example, "All informative things are useful", "Some websites are not useful", therefore "Some websites are not informative". (Contributed by David A. Wheeler, 28-Aug-2016) Reduce dependencies on axioms. (Revised by BJ, 16-Sep-2022)

	Ref	Expression
Hypotheses	baroco.maj	$⊢ \forall x (φ \to ψ)$
	baroco.min	$⊢ \exists x (χ \land \neg ψ)$
Assertion	baroco	$⊢ \exists x (χ \land \neg φ)$

Proof

Step	Hyp	Ref	Expression
1		baroco.maj	$⊢ \forall x (φ \to ψ)$
2		baroco.min	$⊢ \exists x (χ \land \neg ψ)$
3		con3	$⊢ (φ \to ψ) \to (\neg ψ \to \neg φ)$
4	3	anim2d	$⊢ (φ \to ψ) \to ((χ \land \neg ψ) \to (χ \land \neg φ))$
5	4	alimi	$⊢ \forall x (φ \to ψ) \to \forall x ((χ \land \neg ψ) \to (χ \land \neg φ))$
6	1 5	ax-mp	$⊢ \forall x ((χ \land \neg ψ) \to (χ \land \neg φ))$
7		exim	$⊢ \forall x ((χ \land \neg ψ) \to (χ \land \neg φ)) \to (\exists x (χ \land \neg ψ) \to \exists x (χ \land \neg φ))$
8	6 2 7	mp2	$⊢ \exists x (χ \land \neg φ)$

Metamath Proof Explorer

Theorem baroco

Proof